Accuracy of a one-dimensional reduction of dynamical systems on networks

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.